V11 records the six-region theorem, repairs the small-r domain case, and reports an independent coefficient-table reconstruction whose files are not present in this direct intake.
Evidence posture · Reported resultGraph theory · crossing numbers · graph coloring
Albertson Conjecture
Collaboration betaMust every graph have at least as many crossings as the complete graph whose number of vertices equals the graph's chromatic number?
Known results and sources
Research problem
Exact mathematical statement
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Albertson Conjecture stands
The retained V6 foundation remains the reproducible internal baseline at r<=5441 and n<=36726. V11 adds a source-recorded six-region coalescence theorem and reports a live but unreproduced descent to r<=2780 and n<=18764. The former r=5441 strip remains in history rather than the current frontier. The immediate finite task is a complete independent checker over 14,595 orders at r=2780; the principal mathematical obstruction is a five-region state showing that total edge counts alone are insufficient. The direct intake contains only Markdown, so the named code, coefficient table, range ledger, and checker artifacts remain unobserved source assertions.
Choose the color-pair skeleton jointly with paths to reduce crossing weights, balance local degrees, or preserve a stronger residual.
Route status · Active routeThe universal assertion is refuted by all odd balanced C5 blow-ups; retain only conditional paired kernels and capacitated lifting.
Route status · Refuted routeThe retained route narrows a hypothetical globally minimal counterexample to 8<=r<=5441 and n<=36726.
Evidence posture · Reported reductionFor every odd k>=3, the balanced k-fold blow-up of C5 satisfies the retained factor-critical maximal triangle-free boundary conditions but admits no vertex and perfect matching satisfying the paired-router criterion; alternating orientations would have to two-color an odd five-cycle.
Evidence posture · Source-reported route statement · dependencies incompleteRegenerate a complete exact certificate for every claimed post-V8 parameter pair and checker-certify all 14,595 orders at r=2780, with explicit witnesses and exact range counts.
Task status · Work already reported in progressWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Albertson Conjecture in numbers
5.7kretained lines of mathematical investigation5,668 in the current working snapshot
- Argument development
- 4,579 · 81%
- Explored or eliminated routes
- 259 · 5%
- Computational analysis
- 331 · 6%
- Open obligations
- 164 · 3%
- Definitions and setup
- 335 · 6%
25selected mapped statements13routes investigated13reported milestones8open questions6contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Checker-certify the r=2780 frontier
Generate and independently check one exact certificate row for each of the 14,595 admissible orders 4170<=n<=18764 at r=2780.
Suggested move: Build the deterministic generator and checker, preserve every active witness and first failed inequality, and start with the exact six j-intervals recorded by V10.
What would count as progress
- The checker visits all 14,595 admissible orders exactly once.
- Every accepted row has a valid exact witness and margin.
- Uncovered or invalid rows remain explicit and do not lower the cutoff.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
Scroll horizontally to explore the route
Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Choose the color-pair skeleton jointly with paths to reduce crossing weights, balance local degrees, or preserve a stronger residual.
Route status · Active routeUse the small planarization order bounds for r=8 through 15 in a separate structural or bounded-certificate program.
Route status · Active routeBuild a fresh generator and independent checker for the source-reported post-V8 descent, including all 14,595 r=2780 orders, without treating aggregate range totals as coverage evidence.
Route status · Active routePursue degree-profile-sensitive sampling, rigid anticomplete mates, chromatic-plus-residual capacity, or completion on a pruned core; every candidate theorem must pass the recorded adversarial state.
Route status · Active routeUse structural classification or independently checkable SAT, ILP, or exact graph-generation certificates for the small order caps instead of forcing the asymptotic route.
Route status · Active routeExplored alternatives
Other routes
The r=5441-specific six-neighborhood packing target is historical after the source-reported r=2780 descent. Revisit the structural idea only after a complete post-V8 replay identifies an analogous current bottleneck; it is not an active r=5441 work order.
Route status · Route held in reserveThe weighted-routing proposal was calibrated to the superseded r=5441 first strip. Preserve it as a reusable technique, but do not present it as current work unless the reproduced post-V8 certificates expose a comparable route-length deficit at the r=2780 frontier.
Route status · Route held in reserveThe former one-chromatic-value-at-a-time r=5441 program is historical. Resume descent only through a complete certificate/checker reproduction of the post-V8 range and the r=2780 frontier.
Route status · Route held in reserveBrowse 5 more explored routes
The universal assertion is refuted by all odd balanced C5 blow-ups; retain only conditional paired kernels and capacitated lifting.
Route status · Refuted routeThe correct weighted lifting theorem remains a plausible finite-case tool, but it is archived rather than load-bearing in the V6 large-r route.
Route status · Route held in reserveFour-set averaging proves the current one-step tau_4 threshold unattainable at n=2r-1; only materially different deletion or structural variants remain viable.
Route status · Useful but insufficientThe source retains this material in the archive but explicitly rejects it as the primary large-r route after the sharper deterministic certificate.
Route status · Route held in reserveThe retained finite parameter universe is far too broad for naive enumeration; small-r structural or bounded certificates and top-down strip analysis are the scoped alternatives.
Route status · Useful but insufficientRoute statements and reductions
Statements the next route can inspect and build on
Choose one actual edge between every pair of color classes and one artificial path through each class. The selected skeleton P has binomial(r,2) edges and maximum degree at most r-1, the artificial forest T has n-r links and maximum degree at most 2, and contracting the class paths yields K_r as a minor of P+T.
Source-reported route statement · dependencies incompleteEvery globally minimal Albertson counterexample in the retained argument would satisfy 8<=r<=5441 and r+1<=n<=36726.
Source-reported route statement · dependencies incompleteRevision 8 records a closed formula for the bond quantity h(r), exact discrete optimizers for the fixed-order bounds, and low-interval verifier routines intended to replace heuristic search windows.
Source-reported route statement · dependencies incompleteThe current work reports exact-arithmetic sweeps reducing a hypothetical counterexample to 8<=r<=2780 and n<=18764, but it does not retain the complete generator, per-row certificate shards, and independent finite checker required to reproduce the whole descent.
Source-reported route statement · dependencies incompleteA specified normalized five-region state defeats the current total-edge relaxation, so a five-region theorem needs additional structure such as degree profiles, rigid mates, chromatic residual capacity, or a pruned core.
Source-reported route statement · dependencies incompleteThe recorded planarization bounds eliminate r=6,7 and give small order caps for r=8 through 15, which should be handled by a separate structural or independently checkable finite program.
Source-reported route statement · dependencies incompleteMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Checker-certify the r=2780 frontier
Generate and independently check one exact certificate row for each of the 14,595 admissible orders 4170<=n<=18764 at r=2780.
Suggested move: Build the deterministic generator and checker, preserve every active witness and first failed inequality, and start with the exact six j-intervals recorded by V10.Ready to work on
Audit the paired-threshold separator construction
Independently audit the complete six-region coalescence argument, including the separate r=8..11 planarization cases, the a>=2 scope, graph-theoretic hypotheses, endpoint-minimum argument, and coefficient-table reconstruction.
Suggested move: Retrieve the exact coefficient JSON and independent checker named by V11, verify their digests, then review the graph-theoretic proof separately from the algebra.Ready to work on
Choose skeleton and routes jointly
Optimize the selected color-pair skeleton together with routes to lower crossing weights, distribute skeleton degrees, or preserve stronger residual structure.
Suggested move: Test objectives that reduce c_P on frequently used guides or leave a denser residual while preserving a clique-minor completion route.Ready to work on
Rebuild the five-region evaluator
Reproduce the normalized five-region diagnostic with an exact-rational or certified-interval evaluator that records objectives, candidate optimizers, component values, and the final margin.
Suggested move: Implement a standard-library exact evaluator and first reproduce the recorded adversarial normalized state.Ready to work on
Close the small chromatic range structurally
Treat r=8 through 15 using their small planarization order bounds, with structural arguments or bounded reproducible certificates separate from the large-r routing calculation.
Suggested move: Start from the retained integer upper bounds n<=11,19,26,37,47,62,77,96 for r=8,...,15 and seek graph-structural exclusions before any finite search.Ready to work on
Independently certify scaffolded insertion
Independently check the local disk lemma and route-dependent insertion theorem, including endpoint-arcs, erased guides, track concentration, and self-intersection removal.
Suggested move: Reread the local disk and route-dependent proofs against each item in the adversarial audit before relying on them for publication.Ready to work on
Obtain the V11 verifier package
Obtain and digest-bind the code, JSON, manifest, environment, and outputs named in V11 before treating the algebra or range checks as observed evidence.
Suggested move: Request the source package containing every artifact listed in the V11 ledger and compare each byte digest before execution.Blocked by the current route
Reproduce the post-V6 sweep
Regenerate a complete exact certificate for every claimed post-V8 parameter pair and checker-certify all 14,595 orders at r=2780, with explicit witnesses and exact range counts.
Suggested move: Obtain the referenced package artifacts, then generate sorted per-row JSONL certificates and run an independent checker over the complete claimed ranges.Work already reported in progress
Sourced mathematical context
The known mathematical landscape
External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintCranston's preprint verifies r <= 24 and leaves only three (r,|G|) parameter pairs for possible counterexamples with r in {25,26}.[2] Peer reviewedAckerman's improved crossing-lemma bound implied the conjecture for r <= 18.[8] Peer reviewedBarát and Tóth extended the verified range to r <= 16.[7] Peer reviewedAlbertson, Cranston, and Fox proved the conjecture for 7 <= r <= 12; together with earlier cases this verifies r <= 12.[6]
Mathematical neighborhood
Related results and reusable starting points
Current focusAlbertson's conjecture
The r=5 case of Albertson's conjecture is equivalent to the Four Color Theorem.
[6]Several verified ranges are obtained by combining edge lower bounds for critical graphs with crossing-number lower bounds.
[2]Albertson's inequality is benchmarked against cr(K_r); exact values are known only through r=12, while the standard formula is conjectured for all r.
[2]A 2025 SoCG paper uses immersion structure to exclude asymptotic ranges of minimal counterexamples.
[3]Later mathematical changes
What changed after the initial research map
Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.
Six-region theorem and r=2780 frontier replace the r=5441 leadV11 records a six-region coalescence theorem, corrects its small-r domain, and reports an unreproduced descent to r=2780 while preserving the former r=5441 frontier as history.
Albertson V11 prepared date · Changed the research frontierLater mathematical revision
Post-V6 cutoff is separated from reproducible evidenceRevision 8 records a lower provisional cutoff and stronger exact tooling while making clear that the sweep must be regenerated and one earlier multiplier is withdrawn.
Albertson Conjecture comprehensive revised audit V8 date · Changed the research frontierLater mathematical revision
The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.
Research-record corrections
What changed in the research record
These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.
How the route was assembled
Argument structure
These stages follow the mathematical order of the supplied argument.
10 mapped milestonesretained argument map
Browse all 10 mapped stages
- stage 1Global minimal-counterexample framework
- stage 2Color-class skeleton established
- stage 3Diameter bounds sharpened at the critical endpoints
- stage 4Metric color-class path cover added
- stage 5Scaffolded insertion strengthened
- stage 6Edge-disjoint completion certificate established
- stage 7Uniform analytic range excluded
- stage 8Exact bridge excludes r=5442 through 5445
- stage 9Finite conditional universe established
- stage 10First surviving strip reduced to four explicit rows
Global minimal-counterexample frameworkThe retained source reduces a hypothetical failure to a globally minimal counterexample with criticality, degree, connectivity, minor, and order constraints.
Research stage 1Mapped research milestoneInitial research sequence
Color-class skeleton establishedOne selected edge per color pair and one artificial path per color class produce a controlled scaffold contracting to K_r.
Research stage 2Mapped research milestoneInitial research sequence
Diameter bounds sharpened at the critical endpointsThe general minimum-degree diameter stratification is refined to 15 and 16 at the first two points of the j=6 interval.
Research stage 3Mapped research milestoneInitial research sequence
Metric color-class path cover addedDistance-two auxiliary graphs force enough short class-path links to improve total scaffold length throughout every nonempty diameter interval.
Research stage 4Mapped research milestoneInitial research sequence
Scaffolded insertion strengthenedSeparating through-arcs from endpoint-arcs removes unnecessary local crossing charges and supersedes the V5 path-forest error term.
Research stage 5Mapped research milestoneInitial research sequence
Edge-disjoint completion certificate establishedThe sharpened skeleton insertion bound and the dense residual crossing bound combine into one exact deterministic inequality.
Research stage 6Mapped research milestoneInitial research sequence
Uniform analytic range excludedThe source-reported rational endpoint and monotonicity estimates exclude every globally minimal counterexample with r>=5446.
Research stage 7Mapped research milestoneInitial research sequence
Exact bridge excludes r=5442 through 5445The embedded exact-integer loops report positive margins at every allowable order in four chromatic rows below the analytic cutoff.
Research stage 8Mapped research milestoneInitial research sequence
Finite conditional universe establishedPlanarization, the chromatic bridge, and the retained order cap narrow a hypothetical globally minimal counterexample to 8<=r<=5441 and n<=36726.
Research stage 9Mapped research milestoneInitial research sequence
First surviving strip reduced to four explicit rowsAt r=5441 the current certificate leaves four orders, with explicit extra short-link requirements and six-neighborhood rigidity as the primary structural target.
Research stage 10Mapped research milestoneInitial research sequence
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
22 standing statements3 proposed statements13 mathematical milestones8 open questions7 conditional results2 completed special cases
- theorem candidate
1 of 25 1 - reduction
7 of 25 7 - lemma
11 of 25 11 - negative result
3 of 25 3 - computational claim
2 of 25 2 - counterexample
1 of 25 1
Selected mathematical clusters9 mathematical clusters
Conjecture and minimal-counterexample foundationThe exact conjectural inequality and the conditional criticality, degree, connectivity, minor, and order package used by every retained route.3 displayed rows
- retained route statementAlbertson conjecture
- retained route statementGlobal minimal-counterexample packageconditional
- Recorded relationshipThe retained route begins by assuming a counterexample and choosing one globally minimal in (|V|,|E|).reduces to · reported by source
Color skeleton and metric geometryThe one-edge-per-color-pair skeleton, diameter bounds, and distance-two auxiliary path-cover construction.6 displayed rows
- retained route statementColor-pair skeleton and contraction pathsintermediate
- retained route statementMinimum-degree diameter stratificationintermediate
- retained route statementEndpoint-sensitive diameter boundsintermediate
- retained route statementDistance-two packing boundintermediate
- retained route statementMetric color-class path coverintermediate
- DerivationA minimum path cover of each distance-two auxiliary graph uses at most its independence number j paths. Auxiliary-path links have length two and the at most j-1 connectors per color class have length at most the graph diameter, yielding q and L.active reported
Scaffold insertion and residual crossing certificateSharpened local insertion accounting combines with the dense residual to produce the exact deterministic completion inequality.6 displayed rows · 2 routes included
- retained route statementSharpened deterministic scaffolded insertionintermediate
- retained route statementEdge-disjoint support/residual certificateintermediate
- DerivationInsert the class-path forest alongside routes to lower-bound internal skeleton crossings, delete only P to retain a dense residual, apply the crossing lemma there, and add the two edge-pair-disjoint internal crossing counts.active reported
- Research targetIndependently certify scaffolded insertionopen
- Route held in reserveDrawing-weighted routingThe weighted-routing proposal was calibrated to the superseded r=5441 first strip. Preserve it as a reusable technique, but do not present it as current work unless the reproduced post-V8 certificates expose a comparable route-length deficit at the r=2780 frontier.
- Active routeJoint skeleton and route optimizationChoose the color-pair skeleton jointly with paths to reduce crossing weights, balance local degrees, or preserve a stronger residual.
Large-r analytic and exact reductionsThe source-reported uniform argument and finite exact bridge reduce a hypothetical counterexample to r<=5441 and n<=36726.11 displayed rows · 3 routes included
- retained route statementUniform exclusion for r at least 5446conditional
- retained route statementExact bridge exclusion for r=5442 through 5445computational
- retained route statementCurrent finite parameter reductionconditional
- DerivationNormalize n-r by r, upper-bound the insertion error on each of six diameter intervals, and show the exact rational lower bounds for Phi and its derivative are positive at r=5446; monotonicity extends the contradiction to larger r.active reported
- DerivationFor each allowable integer pair with 5442<=r<=5445, the retained program substitutes the sharpened diameter and guide-length bounds into the doubled insertion error and verifies the exact integer numerator N(r,n)>0.active reported
- DerivationThe bridge gives r<=5441, planarization excludes r=6,7 and gives r>=8, and the strict n<27r/4 order bound yields n<=36726 at the largest surviving chromatic value.active reported
- ComputationExact rational evaluation of lower bounds for Phi' and Phi at the left endpoint of each of the six diameter intervals when r=5446.The source reports every derivative lower bound and every endpoint value strictly positive, supporting the uniform analytic exclusion for r>=5446. · reported unreproduced
- ComputationEmbedded exact-integer certificate over every allowable order for r=5442, 5443, 5444, and 5445.The source reports N(r,n)>0 for every integer ceil(3r/2)<=n<27r/4 in those four chromatic rows. · reported unreproduced
- Route held in reserveDescending exact-certificate programThe former one-chromatic-value-at-a-time r=5441 program is historical. Resume descent only through a complete certificate/checker reproduction of the post-V8 range and the r=2780 frontier.
- Active routeSmall-chromatic structural closureUse the small planarization order bounds for r=8 through 15 in a separate structural or bounded-certificate program.
- Useful but insufficientNaive enumeration of the finite universeThe retained finite parameter universe is far too broad for naive enumeration; small-r structural or bounded certificates and top-down strip analysis are the scoped alternatives.
First surviving strip and packing rigidityFour orders at r=5441 remain for the current inequality, with explicit path-merger requirements and an almost-partition by six closed neighborhoods.13 displayed rows · 4 routes included
- supersededFirst surviving strip at r=5441computational
- retained route statementSix-neighborhood packing rigidityconditional
- DerivationRerunning the exact bridge certificate at r=5441 leaves four orders. Decreasing L by (D-2) for each additional distance-two link determines the row-specific counts sufficient to make the inequality strict.active reported
- DerivationSix mutually distance-three vertices have pairwise disjoint closed neighborhoods of at least r+1 vertices each; subtracting 6(r+1) from the four surviving orders leaves only 2 through 5 vertices outside.active reported
- ComputationEmbedded exact arithmetic at r=5441 identifying nonpositive certificate rows and the additional short-link count required in each.The source reports exactly four failing orders, (32654,32655,32656,32657), with guaranteed short-link counts (8,9,10,11) and additional requirements (5,3,2,1). · reported unreproduced
- Research targetClose the four surviving orders at r=5441superseded
- Research targetExploit drawing-weighted routingsuperseded
- Research targetChoose skeleton and routes jointlyopen
- Research targetDescend the finite chromatic rangesuperseded
- Route held in reserveDistance-two packing rigidityThe r=5441-specific six-neighborhood packing target is historical after the source-reported r=2780 descent. Revisit the structural idea only after a complete post-V8 replay identifies an analogous current bottleneck; it is not an active r=5441 work order.
- Route held in reserveDrawing-weighted routingThe weighted-routing proposal was calibrated to the superseded r=5441 first strip. Preserve it as a reusable technique, but do not present it as current work unless the reproduced post-V8 certificates expose a comparable route-length deficit at the r=2780 frontier.
- Active routeJoint skeleton and route optimizationChoose the color-pair skeleton jointly with paths to reduce crossing weights, balance local degrees, or preserve a stronger residual.
- Route held in reserveDescending exact-certificate programThe former one-chromatic-value-at-a-time r=5441 program is historical. Resume descent only through a complete certificate/checker reproduction of the post-V8 range and the r=2780 frontier.
Boundary certificates and exact failuresThe conditional paired-router theorem, its capacitated replacement, and exact counterexamples or averaging barriers that block two universal shortcuts.12 displayed rows · 3 routes included
- retained route statementPaired-router criterionconditional
- retained route statementOdd balanced C5 blow-ups defeat universal paired transversalsspecial case
- retained route statementCapacitated paired-kernel liftingconditional
- retained route statementFour-set averaging upper boundintermediate
- retained route statementOne-step deletion threshold lower boundintermediate
- retained route statementOne-step persistent-edge certificate is too weak at n=2r-1conditional
- DerivationAverage the edge weight incident to a uniformly random four-set to upper-bound tau_4, then compare it with the lower bound 4/r on the one-step deletion threshold.active reported
- Useful failureUniversal paired-transversal certificate at the n=2r-1 boundaryreported failure
- Useful failureOne-step persistent-edge certificate at n=2r-1reported failure
- Refuted routeUniversal paired-transversal certificateThe universal assertion is refuted by all odd balanced C5 blow-ups; retain only conditional paired kernels and capacitated lifting.
- Route held in reserveCapacitated paired-kernel finite casesThe correct weighted lifting theorem remains a plausible finite-case tool, but it is archived rather than load-bearing in the V6 large-r route.
- Useful but insufficientOne-step persistent-edge certificateFour-set averaging proves the current one-step tau_4 threshold unattainable at n=2r-1; only materially different deletion or structural variants remain viable.
Active, paused, and too-weak routesThe source prioritizes packing rigidity, weighted routing, skeleton optimization, descending exact certificates, and small-r closure while pausing archived stochastic routes and rejecting naive enumeration.12 displayed rows · 7 routes included
- Research targetClose the four surviving orders at r=5441superseded
- Research targetExploit drawing-weighted routingsuperseded
- Research targetChoose skeleton and routes jointlyopen
- Research targetDescend the finite chromatic rangesuperseded
- Research targetClose the small chromatic range structurallyopen
- Route held in reserveDistance-two packing rigidityThe r=5441-specific six-neighborhood packing target is historical after the source-reported r=2780 descent. Revisit the structural idea only after a complete post-V8 replay identifies an analogous current bottleneck; it is not an active r=5441 work order.
- Route held in reserveDrawing-weighted routingThe weighted-routing proposal was calibrated to the superseded r=5441 first strip. Preserve it as a reusable technique, but do not present it as current work unless the reproduced post-V8 certificates expose a comparable route-length deficit at the r=2780 frontier.
- Active routeJoint skeleton and route optimizationChoose the color-pair skeleton jointly with paths to reduce crossing weights, balance local degrees, or preserve a stronger residual.
- Route held in reserveDescending exact-certificate programThe former one-chromatic-value-at-a-time r=5441 program is historical. Resume descent only through a complete certificate/checker reproduction of the post-V8 range and the r=2780 frontier.
- Active routeSmall-chromatic structural closureUse the small planarization order bounds for r=8 through 15 in a separate structural or bounded-certificate program.
- Route held in reserveConductance and random-walk routingThe source retains this material in the archive but explicitly rejects it as the primary large-r route after the sharper deterministic certificate.
- Useful but insufficientNaive enumeration of the finite universeThe retained finite parameter universe is far too broad for naive enumeration; small-r structural or bounded certificates and top-down strip analysis are the scoped alternatives.
Revision-8 reproducibility frontierThe reported lower cutoff, exact optimizer tools, withdrawn multiplier, and two immediate reproduction/audit obligations.6 displayed rows · 1 route included
- supersededReported post-V6 cutoff requiring reproductioncomputational
- retained route statementExact fixed-order optimizer toolkitintermediate
- Useful failureFirst multiball completion coefficient used in the attempted r<=3043 extensionreported failure
- Research targetReproduce the post-V6 sweepin progress reported
- Research targetAudit the paired-threshold separator constructionopen
- Active routeCertificate-first post-V6 reproductionBuild a fresh generator and independent checker for the source-reported post-V8 descent, including all 14,595 r=2780 orders, without treating aggregate range totals as coverage evidence.
V11 six-region and r=2780 frontierThe V11 source revision combines the six-region structural advance, a lower unreproduced finite frontier, exact evidence boundaries, and the five-region route portfolio.13 displayed rows · 3 routes included
- retained route statementSix-region coalescenceintermediate
- retained route statementClosed paired-threshold capintermediate
- retained route statementReported post-V8 cutoff requiring reproductioncomputational
- retained route statementFive-region total-edge relaxation is insufficientintermediate
- retained route statementSeparate bounded small-chromatic programspecial case
- Research targetChecker-certify the r=2780 frontieropen
- Research targetObtain the V11 verifier packageblocked
- Research targetRebuild the five-region evaluatoropen
- ComputationSource-reported independent reconstruction of ten endpoint brackets and twelve endpoint-sum margins for the six-region theorem.V11 reports coefficient-for-coefficient agreement and strict positivity, but the direct Markdown intake does not contain the checker or coefficient JSON, so ProofAtlas did not rerun it. · reported unreproduced
- ComputationSource-reported arithmetic ledger for 1,743,870 parameter pairs in the post-V8 finite ranges.V11 reports matching pair totals and positive weakest margins, while explicitly denying that the range ledger establishes complete row coverage or independently checked witnesses. · reported unreproduced
- Active routeCertificate-first post-V6 reproductionBuild a fresh generator and independent checker for the source-reported post-V8 descent, including all 14,595 r=2780 orders, without treating aggregate range totals as coverage evidence.
- Active routeFive-region structure beyond total edge countsPursue degree-profile-sensitive sampling, rigid anticomplete mates, chromatic-plus-residual capacity, or completion on a pruned core; every candidate theorem must pass the recorded adversarial state.
- Active routeBounded small-chromatic closureUse structural classification or independently checkable SAT, ILP, or exact graph-generation certificates for the small order caps instead of forcing the asymptotic route.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
The current research map records this as an open mathematical step.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- Every allowable post-V8 parameter pair is represented exactly once by a machine-readable certificate row.
- Every witness and exact margin passes an independent checker.
- All 14,595 orders at r=2780 are either certified or recorded as explicit uncovered rows; no lower cutoff is claimed from isolated rows.
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Prepared starting pointChecker-certify the r=2780 frontier
Albertson Conjecture · ready to start
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Research contextPrepared context for any AI agent
Must every graph have at least as many crossings as the complete graph whose number of vertices equals the graph's chromatic number?
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Sources and references8 cited works · next context review by Nov 2, 2026
The mathematical context was checked on Aug 2, 2026. Status can be refreshed sooner after a material result or claim.
- 1ARCC Workshop: Albertson conjecture and related problemsauthoritative webpage · accessed Aug 2, 2026
- 2Progress on Albertson's Conjecturepreprint · accessed Aug 2, 2026
- 3https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.50authoritative webpage · accessed Aug 2, 2026
- 4Dedicated English Wikipedia articleencyclopedia · accessed Aug 2, 2026
- 5Crossings, Colorings, and Cliquesauthoritative webpage · accessed Aug 2, 2026
- 6Crossings, Colorings, and Cliquesoriginal source · accessed Aug 2, 2026
- 7Towards the Albertson Conjectureauthoritative webpage · accessed Aug 2, 2026
- 8On topological graphs with at most four crossings per edgeauthoritative webpage · accessed Aug 2, 2026
Important qualifications
- The r <= 24 frontier is sourced to a December 2025 preprint and is therefore labeled as preprint progress rather than silently presented as a peer-reviewed theorem.
- No authoritative theorem-level formalization or maintained computational benchmark was identified in the scoped search; empty resource lists are not assertions of nonexistence.
- Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.
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